Problem: You have found the following ages (in years) of all 5 sloths at your local zoo: $ 3,\enspace 12,\enspace 18,\enspace 12,\enspace 19$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Answer: Because we have data for all 5 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $5$ ages and divide by $5$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5}} $ $ {\mu} = \dfrac{3 + 12 + 18 + 12 + 19}{{5}} = {12.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $3$ years $-9.8$ years $96.04$ years $^2$ $12$ years $-0.8$ years $0.64$ years $^2$ $18$ years $5.2$ years $27.04$ years $^2$ $12$ years $-0.8$ years $0.64$ years $^2$ $19$ years $6.2$ years $38.44$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{96.04} + {0.64} + {27.04} + {0.64} + {38.44}} {{5}} $ $ {\sigma^2} = \dfrac{{162.8}}{{5}} = {32.56\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{32.56\text{ years}^2}} = {5.7\text{ years}} $ The average sloth at the zoo is 12.8 years old. There is a standard deviation of 5.7 years.